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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > reprle | Structured version Visualization version GIF version |
Description: Upper bound to the terms in the representations of 𝑀 as the sum of 𝑆 nonnegative integers from set 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
Ref | Expression |
---|---|
reprval.a | ⊢ (𝜑 → 𝐴 ⊆ ℕ) |
reprval.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
reprval.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
reprf.c | ⊢ (𝜑 → 𝐶 ∈ (𝐴(repr‘𝑆)𝑀)) |
reprle.x | ⊢ (𝜑 → 𝑋 ∈ (0..^𝑆)) |
Ref | Expression |
---|---|
reprle | ⊢ (𝜑 → (𝐶‘𝑋) ≤ 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6901 | . 2 ⊢ (𝑎 = 𝑋 → (𝐶‘𝑎) = (𝐶‘𝑋)) | |
2 | fzofi 13994 | . . 3 ⊢ (0..^𝑆) ∈ Fin | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (0..^𝑆) ∈ Fin) |
4 | reprval.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
5 | reprval.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | reprval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
7 | reprf.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴(repr‘𝑆)𝑀)) | |
8 | 4, 5, 6, 7 | reprsum 34459 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎) = 𝑀) |
9 | 4 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ) |
10 | 4, 5, 6, 7 | reprf 34458 | . . . . 5 ⊢ (𝜑 → 𝐶:(0..^𝑆)⟶𝐴) |
11 | 10 | ffvelcdmda 7098 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (𝐶‘𝑎) ∈ 𝐴) |
12 | 9, 11 | sseldd 3980 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (𝐶‘𝑎) ∈ ℕ) |
13 | 12 | nnrpd 13068 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (𝐶‘𝑎) ∈ ℝ+) |
14 | reprle.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (0..^𝑆)) | |
15 | 1, 3, 8, 13, 14 | fsumub 32729 | 1 ⊢ (𝜑 → (𝐶‘𝑋) ≤ 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 ⊆ wss 3947 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 Fincfn 8974 0cc0 11158 ≤ cle 11299 ℕcn 12264 ℕ0cn0 12524 ℤcz 12610 ..^cfzo 13681 reprcrepr 34454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-ico 13384 df-fz 13539 df-fzo 13682 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-sum 15691 df-repr 34455 |
This theorem is referenced by: hgt750lemb 34502 |
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